Comparing alternatives using multiple criteria

Transcription

1 Comparing alternatives using multiple criteria Denns L. Bricker Dept of Mechanical & Industrial Engineering The University of Iowa AHP 2/4/2003 page 1 of 22

2 When a decision-maker has multiple objectives, it is difficult to choose between alternatives, e.g. Choosing which of several job offers to accept o Salary o Location o Opportunity for advancement o Personal interests Selecting which automobile to purchase o Price o Safety o Fuel economy o Power of engine o Style o Reliability AHP 2/4/2003 page 2 of 22

3 Selecting which university to attend What are some of the criteria that you used?,, Selecting plant site What do you think are some of the criteria that a company uses?,, AHP 2/4/2003 page 3 of 22

4 EXAMPLE You are trying to decide whether to live in Chicago or New York City, based upon four criteria: 1. Housing cost 2. Cultural opportunities 3. Quality of schools 4. Crime level How do you decide which city to choose? AHP 2/4/2003 page 4 of 22

5 First you need to determine the relative importance of the criteria. for example, housing cost might be more important than cultural opportunities! These might be specified by a weight assigned to each criterion: Weights for criteria: Criterion Housing cost Culture Schools Crime Weight By convention, these weights have been scaled so that the sum is 1.00, but this isn t really necessary! AHP 2/4/2003 page 5 of 22

6 Weights for criteria: Criterion Housing cost Culture Schools Crime Weight These weights reflect a subjective judgment that housing cost is 5 times as important as cultural opportunities, while crime rate is only twice as important, etc. AHP 2/4/2003 page 6 of 22

7 Next you need to compare the two cities using each criterion: (Given proper data, this might be done objectively in this particular instance, but sometimes subjective judgment is required.) Housing cost Culture Schools Crime New York Chicago (Each pair of scores must sum to 100%!) According to these scores, Chicago ranks better in housing cost, i.e., lower costs, and worse in cultural opportunities, i.e., less AHP 2/4/2003 page 7 of 22

8 Based upon these values, you can now compute a score for each city: weight of score of city weight of score of city housing w.r.t. housing + + crime rate w.r.t. crime rate New York: = 0.4 Chicago: = 0.6 Accordingly, your better choice would seem to be Chicago! AHP 2/4/2003 page 8 of 22

9 T. Saaty proposed AHP as a method for systematically determining the weights of the criteria & of the cities with respect to the criteria. by using pair-wise comparisons. AHP 2/4/2003 page 9 of 22

10 The standard AHP approach assumes that when criteria i & j are compared, the following rating is assigned to the more preferred one: Rating Description 1 Equally preferred/important 3 Moderately preferred/important 5 Strongly preferred/important 7 Very strongly preferred/important 9 Extremely strongly preferred/important Ratings 2, 4, 6 & 8 may be used as well, with obvious interpretations. AHP 2/4/2003 page 10 of 22

11 A square table is created with the ratings-- for example, with three alternatives being compared: Alternative #1 Alternative #2 Alternative #3 Alternative #1 1 5 Alternative # Alternative #3 1 If Alternative #1 is strongly preferred to Alternative #2, then we enter 5 into row #1 & column #2. At the same time, we enter its reciprocal, 1 5, in row #2 & column #1. Along the diagonal, we always enter the value 1. AHP 2/4/2003 page 11 of 22

12 For example: Housing cost Cultural opportunities Housing 1 5 cost Cultural opportunities School quality Crime rate School quality 1 Crime rate 1 For example, Housing cost is strongly more important than Cultural Opportunity, as indicated by 5. AHP 2/4/2003 page 12 of 22

13 (developed by Thomas Saaty) 1. Develop the weights for each criterion by a. Developing a pairwise comparison matrix b. Computing eigenvector of the matrix c. Checking consistency 2. Develop the weights for each alternative with respect to each criterion by a. Developing a pairwise comparison matrix b. Computing eigenvector of the matrix c. Checking consistency 3. Calculate the weighted average for each alternative Choose the alternative yielding the highest score. AHP 2/4/2003 page 13 of 22

14 Alternative determination of weights T. Saaty suggests, for theoretical reasons, using eigenvectors for the weights. A simpler averaging scheme yields a set of weights which are approximately those obtained by the eigenanalysis: Example: A = Column sums are 3.333, 4.2, and 6.5 AHP 2/4/2003 page 14 of 22

15 The normalized matrix is computed by dividing each entry by the column sum: A = = AHP 2/4/2003 page 15 of 22

16 Now we compute the average of the columns: w = = = If we had used eigenanalysis, we would have found the eigenvector The results differ, but not greatly! AHP 2/4/2003 page 16 of 22

17 Example In deciding where to invest your money, you consider two criteria: expected rate of return degree of risk You have decided that rate of return is twice as important to you as avoiding risk, i.e., you assign weights 2 and 1 to the criteria. You are aware of three investment opportunities, for which you have done a pair-wise comparison with each criterion: return #1 #2 #3 risk #1 #2 #3 #1 1 1/3 4 # /2 # #2 1/4 1 1/6 #3 1/4 1/8 1 # How should these investments be ranked? AHP 2/4/2003 page 17 of 22

18 Compute the sum of each column: return #1 #2 #3 risk #1 #2 #3 #1 1 1/3 4 # /2 # #2 1/4 1 1/6 #3 1/4 1/8 1 # Sum: Sum: Normalize each column so that entries in the column sum to 1, by dividing each entry by the sum: return #1 #2 #3 risk #1 #2 #3 # # # # # # Sum: Sum: AHP 2/4/2003 page 18 of 22

19 Now compute the average of each row of the two matrices return #1 #2 #3 Average # # # risk #1 #2 #3 Average # # # These averages are therefore the ratings for the investments with respect to each criterion. AHP 2/4/2003 page 19 of 22

20 Computation of final score for the investments: Rate of return Degree of risk Total score Criterion weight 2 1 Investment # Investment # Investment # Investment #2 receives the highest overall rating, and investment #1 is rated slightly higher than #3. AHP 2/4/2003 page 20 of 22

21 Computation of weights, using Excel AHP 2/4/2003 page 21 of 22

22 AHP 2/4/2003 page 22 of 22